Question: Let $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $r$ be a real number.  Two of the roots of $f(x)$ are $r + 1$ and $r + 7.$  Two of the roots of $g(x)$ are $r + 3$ and $r + 9,$ and
\[f(x) - g(x) = r\]for all real numbers $x.$  Find $r.$
Solution: By Factor Theorem,
\[f(x) = (x - r - 1)(x - r - 7)(x - a)\]and
\[g(x) = (x - r - 3)(x - r - 9)(x - b)\]for some real numbers $a$ and $b.$

Then
\[f(x) - g(x) = (x - r - 1)(x - r - 7)(x - a) - (x - r - 3)(x - r - 9)(x - b) = r\]for all $x.$

Setting $x = r + 3,$ we get
\[(2)(-4)(r + 3 - a) = r.\]Setting $x = r + 9,$ we get
\[(8)(2)(r + 9 - a) = r.\]Then $-8r - 24 + 8a = r$ and $16r + 144 - 16a = r,$ so
\begin{align*}
8a - 9r &= 24, \\
-16a + 15r &= -144.
\end{align*}Solving, we find $r = \boxed{32}.$